The mathematics of classical mechanics Newton’s Laws of motion: 1) inertia 2) F = ma 3) action:reaction The motion of particle is represented as a differential equation F = ma = m(dv/dt) = dp/dt Lagrange: Defines the motion of particle in terms of kinetic and potential energy rather than forces. Introduces partial differentials representing each degree of freedom of motion (3D motion) Hamilton: E = K + V — In QM the energy of a particle is determined mathematically by applying the Hamiltonian operator to the wave function representing the particle.
Get Comfortable with units (SI = kg, m, s) Force = Pressure = Energy =
Get Comfortable with units (SI = kg, m, s) Force = m (kg) a (m s -2 ) = kg m s -2 Pressure = F/area = kg m s -2 m 2 = kg m -1 s -2 Energy = F distance = kg m 2 s -2 or PV = kg m -1 s -2 m 3 = kg m 2 s -2 Power = E/t = W (watts) = J s -1 = kg m 2 s -3
classical mechanics vs. quantum mechanics Particle – wave distinction Energy is continuous Deterministic Particle – wave duality uncertainty principle Energy is quantized
Light (emr) – James C. Maxwell (1831 – 1879) CM: 1) emr is a wave form of energy 2) travels through ‘ether’ 3) E is continuous 4) wave amplitude = intensity QM: 1) Photons of light (Relativity ‘removed’ ether) 2) Energy is quantized: E = h 3) Intensity = # photons rays x rays UV IR waves radio
CLASSICAL MECHANICS failures Atomic structure and spectra: Predicts collapse of electrons into nucleus of atoms No explanation for spectrum of atom Blackbody Radiation: Predicts a heated body will emit infinite energy The Photoelectric Effect Predicts that ↑intensity of emr should be sufficient to expel e - The Equipartition Theory C P of a substance = ½kT for each degree of freedom C P is independent of temperature
Blackbody Radiation A heated object will emit radiation
CM d = 8 kT/ 4 d CM predicts infinite energy would be emitted by a blackbody! E/V (energy density J m -3 ) slope = d d QM d = {8 hc/ 5 1/{exp(hc/ kT)-1} d
QM predicts a peak of energy density occurring at higher frequencies (lower ) as T↑. This is what is observed. Total power flux (W m -2 or J s -1 m -2 ) = T 4 = x W m -2 K -4 max (nm) = 2.90 x 10 6 (nm K) ÷ T (K) Sun 5780 K Fe 1811 K NaCl 1074 K Lava~ 1200 K Human 310 K CMBR 2.73K W bulb 3000K dd
Sun Visible range
The Photoelectric Effect E = h = h o + ½mv 2 Light striking a solid metal surface may result in e - expulsion. This is not the same as the 1 st ionization potential which is for gas phase CM – High intensity, low light should be able to cause e - to be emitted! QM – intensity # of e - expelled (but 1 photon required for each e - ) Threshold o below which no e - emitted regardless of intensity. E > h o imparts extra speed to emitted e -. minimum E to expel e - Kinetic energy of e - metal work function (eV) Na2.36 K2.9 Cs2.14 Mg3.66 Ca2.87 Mn4.1 Nd3.2 Ag4.6 Sn4.42 Pb4.25
Heat Capacity - Oscillations of a solid CM: C V = 3R (constant for all T) QM: C V = 3R at high T - as T 0; C V 0 Although the equipartition theorem makes very accurate predictions in certain conditions, it becomes inaccurate when quantum effects are significant, such as at low temperatures. When the thermal energy kT is smaller than the quantum energy spacing in a particular degree of freedom, the average energy and heat capacity of this degree of freedom are less than the values predicted by equipartition.quantum effectsdegree of freedom Einstein-Bose condensation
Heat Capacity - Oscillations of a solid CM: C V = 3R (constant for all T) QM: C V = 3R at high T - as T 0; C V 0 where E kT Although the equipartition theorem makes very accurate predictions in certain conditions, it becomes inaccurate when quantum effects are significant, such as at low temperatures. When the thermal energy kT is smaller than the quantum energy spacing in a particular degree of freedom, the average energy and heat capacity of this degree of freedom are less than the values predicted by equipartition.quantum effectsdegree of freedom Einstein-Bose condensation
CM predicts that the electron should radiate energy as it orbits the Nucleus and thus eventually collapse into the nucleus. Observation obviously counters that outcome. ● Atomic Theory: Dalton (1808) – indivisible sphere Thomson (1890s) raisin pudding Rutherford (1908) nuclear model ● emr emitted
de Broglie E = h & E = mc 2 replace particle velocity for c (derive expression for ) mv 2 = hv/ = h/mv h = hc/ = mc 2 in 1927 Ni crystal observed to diffract e - beam 1 st observation of wave properties of ‘particle’
The Bohr Atom - Postulates Energy is Quantized Atoms in stationary state will not emit radiation An atom absorbs or emits radiation as it changes state The Bohr Atom Orbit has radius such that angular momentum, L = mvr = nħ ● The energy of the orbits are quantized (eq. 9.31) 0.529Å